`Bandits.UCB.regret_le'`🔗

Lemma

From the authors

Regret bound for the UCB algorithm with exploration constant c, for σ2-subgaussian rewards.

Types

Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.

Given

K : ℕNeZero K
c : ℝ
ν : ProbabilityTheory.Kernel (Fin K) ℝA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel νA kernel is a Markov kernel if every measure in its image is a probability measure.
P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsProbabilityMeasure PA measure μ is called a probability measure if μ univ = 1.
O : ℕ → Ω → Unit
A : ℕ → Ω → Fin K
R : ℕ → Ω → ℝ
σ2 : NNReal
n : ℕ

Assuming

h : Learning.IsAlgEnvSeq O A R (ucbAlgorithm K c) (Learning.stationaryEnv ν) PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.
hν : ∀ (a : Fin K), ProbabilityTheory.HasSubgaussianMGF (fun x => x - ∫ (x : ℝ), id x ∂ν a) σ2 (ν a)A random variable X has a sub-Gaussian moment-generating function with parameter c with respect to a measure μ if for all t : ℝ, exp (t * X) is μ-integrable and the moment-generating funct…
hσ2 : σ2 ≠ 0
hc : 0 < c

Then

∫ (x : Ω), regret ν A n x ∂P ≤ ∑ a, (8 * c * Real.log (↑n + 1) / gap ν a + gap ν a * (2 + 2 * constSum (c / ↑σ2) n))

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

NeZero : {R : Type u_1} → [Zero R] → R → Prop

A type-class version of `n ≠ 0`.

Nat : Type

The natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.

Real : Type

The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers.

ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A kernel is a Markov kernel if every measure in its image is a probability measure.

ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)

A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.

Fin : ℕ → Type

Natural numbers less than some upper bound.

In particular, a `Fin n` is a natural number `i` with the constraint that `i < n`. It is the
canonical type with `n` elements.

MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called a probability measure if `μ univ = 1`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Unit : Type

The canonical type with one element. This element is written `()`.

`Unit` has a number of uses:
* It can be used to model control flow that returns from a function call without providing other
  information.
* Monadic actions that return `Unit` have side effects without computing values.
* In polymorphic types, it can be used to indicate that no data is to be stored in a particular
  field.

NNReal : Type

Nonnegative real numbers, denoted as `ℝ≥0` within the NNReal namespace

Learning.IsAlgEnvSeq : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {Ω : Type u_4} →
        {m𝓞 : MeasurableSpace 𝓞} →
          {m𝓐 : MeasurableSpace 𝓐} →
            {m𝓨 : MeasurableSpace 𝓨} →
              {mΩ : MeasurableSpace Ω} →
                (ℕ → Ω → 𝓞) →
                  (ℕ → Ω → 𝓐) →
                    (ℕ → Ω → 𝓨) →
                      Learning.Algorithm 𝓞 𝓐 𝓨 →…

An algorithm-environment sequence: a sequence of observations, actions and feedbacks generated
by an algorithm interacting with an environment.

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Bandits.ucbAlgorithm : (K : ℕ) → [NeZero K] → ℝ → Learning.Algorithm Unit (Fin K) ℝ

The UCB algorithm.

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Learning.stationaryEnv : {𝓐 : Type u_2} →
  {𝓨 : Type u_3} →
    {m𝓐 : MeasurableSpace 𝓐} →
      {m𝓨 : MeasurableSpace 𝓨} →
        (ν : ProbabilityTheory.Kernel 𝓐 𝓨) → [ProbabilityTheory.IsMarkovKernel ν] → Learning.Environment Unit 𝓐 𝓨

A stationary environment without observations, in which the distribution of the next feedback
depends only on the last action.

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ProbabilityTheory.HasSubgaussianMGF : {Ω : Type u_1} →
  {mΩ : MeasurableSpace Ω} →
    (Ω → ℝ) → NNReal → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasSubgaussianMGF._auto_1 → Prop

A random variable `X` has a sub-Gaussian moment-generating function with parameter `c`
with respect to a measure `μ` if for all `t : ℝ`, `exp (t * X)` is `μ`-integrable and
the moment-generating function of `X` is bounded by `exp (c * t ^ 2 / 2)` for all `t : ℝ`.
This implies in particular that `X` has expectation 0.

This is equivalent to `Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())`,
as proved in `HasSubgaussianMGF_iff_kernel`.
Properties about sub-Gaussian moment-generating functions should be proved first for
`Kernel.HasSubgaussianMGF` when possible.

HSub.hSub : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HSub α β γ] → α → β → γ

`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).

MeasureTheory.integral : {α : Type u_6} →
  {G : Type u_7} →
    [inst : NormedAddCommGroup G] → [NormedSpace ℝ G] → {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → G) → G

The Bochner integral

id : {α : Sort u} → α → α

The identity function. `id` takes an implicit argument `α : Sort u`
(a type in any universe), and an argument `a : α`, and returns `a`.

Although this may look like a useless function, one application of the identity
function is to explicitly put a type on an expression. If `e` has type `T`,
and `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean
knows that this expression has type `T'` rather than `T`. This can make a
difference for typeclass inference, since `T` and `T'` may have different
typeclass instances on them. `show T' from e` is sugar for an `@id T' e`
expression.

Ne : {α : Sort u} → α → α → Prop

`a ≠ b`, or `Ne a b` is defined as `¬ (a = b)` or `a = b → False`,
and asserts that `a` and `b` are not equal.


Conventions for notations in identifiers:

 * The recommended spelling of `≠` in identifiers is `ne`.

LT.lt : {α : Type u} → [self : LT α] → α → α → Prop

The less-than relation: `x < y` 

Conventions for notations in identifiers:

 * The recommended spelling of `<` in identifiers is `lt`.

LE.le : {α : Type u} → [self : LE α] → α → α → Prop

The less-equal relation: `x ≤ y` 

Conventions for notations in identifiers:

 * The recommended spelling of `≤` in identifiers is `le`.

Bandits.regret : {𝓐 : Type u_2} → {Ω : Type u_3} → {m𝓐 : MeasurableSpace 𝓐} → ProbabilityTheory.Kernel 𝓐 ℝ → (ℕ → Ω → 𝓐) → ℕ → Ω → ℝ

Regret of a sequence of pulls `k : ℕ → 𝓐` at time `t` for the reward kernel `ν ; Kernel 𝓐 ℝ`.

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HMul.hMul : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HMul α β γ] → α → β → γ

`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.

Real.log : ℝ → ℝ

The real logarithm function, equal to the inverse of the exponential for `x > 0`,
to `log |x|` for `x < 0`, and to `0` for `0`. We use this unconventional extension to
`(-∞, 0]` as it gives the formula `log (x * y) = log x + log y` for all nonzero `x` and `y`, and
the derivative of `log` is `1/x` away from `0`.

HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ

`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.

HDiv.hDiv : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HDiv α β γ] → α → β → γ

`a / b` computes the result of dividing `a` by `b`.
The meaning of this notation is type-dependent.
* For most types like `Nat`, `Int`, `Rat`, `Real`, `a / 0` is defined to be `0`.
* For `Nat`, `a / b` rounds downwards.
* For `Int`, `a / b` rounds downwards if `b` is positive or upwards if `b` is negative.
  It is implemented as `Int.ediv`, the unique function satisfying
  `a % b + b * (a / b) = a` and `0 ≤ a % b < natAbs b` for `b ≠ 0`.
  Other rounding conventions are available using the functions
  `Int.fdiv` (floor rounding) and `Int.tdiv` (truncation rounding).
* For `Float`, `a / 0` follows the IEEE 754 semantics for division,
  usually resulting in `inf` or `nan`. 

Conventions for notations in identifiers:

 * The recommended spelling of `/` in identifiers is `div`.

Bandits.gap : {𝓐 : Type u_2} → {m𝓐 : MeasurableSpace 𝓐} → ProbabilityTheory.Kernel 𝓐 ℝ → 𝓐 → ℝ

Gap of an action `a`: difference between the highest mean of the actions and the mean of `a`.

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Bandits.UCB.constSum : ℝ → ℕ → ℝ

A sum that appears in the UCB regret upper bound. For `c > 2` it is bounded uniformly in `n`,
see `constSum_le`.

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Code

lemma regret_le' (h : IsAlgEnvSeq O A R (ucbAlgorithm K c) (stationaryEnv ν) P)
    (hν : ∀ a, HasSubgaussianMGF (fun x ↦ x - (ν a)[id]) σ2 (ν a))
    (hσ2 : σ2 ≠ 0) (hc : 0 < c) (n : ℕ) :
    P[regret ν A n] ≤
      ∑ a, (8 * c * log (n + 1) / gap ν a + gap ν a * (2 + 2 * constSum (c / σ2) n))

Proof

by
  have hσ2' : (0 : ℝ) < σ2 := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr hσ2)
  have h' : IsAlgEnvSeq O A R (ucbAlgorithm K (c / σ2 * σ2)) (stationaryEnv ν) P := by
    rwa [div_mul_cancel₀ _ hσ2'.ne']
  refine (regret_le h' hν hσ2 (div_pos hc hσ2') n).trans_eq ?_
  rw [show (8 : ℝ) * (c / σ2) * σ2 = 8 * c by rw [mul_assoc, div_mul_cancel₀ _ hσ2'.ne']]

New in v4.34.0-rc2-90-gdde3322 (2026-09-16), and its meaning has not changed since.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 24 project declarations, 128 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Bandits.UCB.regret_le'🔗

`Bandits.UCB.regret_le'`🔗